Yes! That happened to me last year when they wanted to start an IV. I kept insisting I needed my right hand that afternoon for my writing. I begged them to please, please put it in the arm, just below the hand. I knew this from past experiences with IVs. Two of them then pounced on me. At that point, I needed the test done and gave in. My hand was swollen for two days. Typing was hurtful. At least no permanent damage.
Next time don't back down.
I have a vein in my right hand that had an IV in it ten years ago. This year, for the very first time, it has started to ache in that precise spot, and the vein is starting to bulge upward in a funny vaguely dark lump. So maybe nothing bad has happened YET to that vein of yours, but it might one day.
Ever seen the hands of really old people? With huge, swollen veins? Some of those swellings they suffer from are IV-induced from years of medical procedures. And as they get odler, elderly people just need more and more medical treatment all the time. Their hands are just horrible looking!
And once again, it's because it's more convenient for the nurse to get their rounds done more quickly. Or else because the surgeon stipulates he wants to be operating on you from X-side of your unconcscious body, and he doesn't want some annoying IV tube, and it's accompanying hanging bag, in his way all during surgery. So the nurses are sometimes being pressured to coerce patients into agreeing to the location of the IV for the surgeon's convenience.
People who use manual wheel chairs (as opposed to power chairs) also have to put up a fight sometimes, because if they were to suffer impairment to one of their hands, they would not be able to operate their own wheelchairs. I know an elderly man in a wheelchair who said his protest escalated into an argument with a nurse one day when he had to get some routine testing done, and he told her if she stuck anything in his hand or wrist he would sue the hospital. She had to get her supervisor and the supervisor agreed with the man in the wheelchair.
YOU HAVE A RIGHT TO REFUSE! Exercise it. Don't back down. Not when it comes to this.